Where does infinite begin?
It was at this point that the stranger said, "Look at the illustration closely. You'll never see it again".
Jorge Luis Borges2
The Book of Sand
The last of the numbers ... let's see how can we find it, because surely, after it is where the infinity begin.
Overall, in mathematics there are so many surprises, that perhaps the largest number is not so far as we imagine, and the same time, the infinite may be closer than expected.
The natural numbers
To begin, we must remember that in the field of mathematics it is always required to define the concepts previously to avoid that some people imagine a few things and others are seeing something similar, but not necessarily the same. Let us start with natural numbers;
Definition: natural number. We call natural number any of the elements of the series 1, 2, 3, 4, ... etc.
It is not hard to realize that the natural numbers are the same that we use every day to count objects and quantities: a book has 100 pages, has a square has 4 corners, etc..
For the purpose of this article, since the zero number is not uses to count, is not classified as a natural number.
As our ancestors began counting their cattle, their sheep, their children and their friends and enemies, early mathematics began with the number 1 –the first natural number. One (1) is the smallest we can count: there is one sun in our solar system, one moon around our planet.
The number two (2) is also a natural numbers, but it always appears when the number one takes the lead in counting. For that reason, the number two is called the successor of the number 1, and in turn, 3 is the successor of 2, etc.. Note that in this case if the 3 is the successor of 2, which in turn is the successor of 1, then 3 is the successor of the successor of 1.
This property of successiveness can be projected indefinitely because if the number 1 has it, and if the number 2 has it, then there is no reason to stop at some point no matter the natural number we choose.
We can use the symbol ( ') to denote the successor of a natural number, in this way we can say that 2 = 1' (i.e., 2 is the successor of 1), 3 = 2 '= 1'', etc.. Therefore, the natural numbers are this:
1, 1 ', 1'', 1''', ...
What is interesting about this mathematical notation is that we do not need the Indo-Arabic numerals (the 10 symbols 1, 2, 3, ... 9) to represent the natural numbers. But eventually, this notation ( ','',''', etc..) will becomes unmanageable, and we must again recourse to the decimal notation system of numbers that we know.
Is there a natural number that has no natural successor?
No matter the notation to which we stick, the property of the successiveness will always be present because we intuitively imagine it like this. Note that it is the property of the successiveness that makes the number: the concept of number arises from the succession of the number one, because the number 1 alone by itself is not a number. If plurality were nonexistent, that is, if of every thing there were a unity, we would not have invented the counting system, and therefore, the mathematics.
Once we intuitively accept the notion of succession, then we have started to accept the existence of infinity, because infinity is this: the unstoppable, the unlimited. It should be emphasized that with this we are not identifying the unreachable with the infinity, because infinity is not something that is never reached, but on the contrary, that never stops.
The Webster Dictionary defines infinite in this way:
Infinite
* 1. extending indefinitely: endless.
* 2. immeasurably or inconceivably great or extensive Inexhaustible.
* 3. subject to no limitation or external determination.
In view of the above, we will do our very simple definition of infinity, but as an abstract concept that we will have to resort by opposition.
Infinite: That has no end, that there's no way to assign a limit, endless.
Where does infinity begin?
The series of natural numbers cannot end because there will always be a number that will be the successor to anyone we accept as the greatest of them by the property of succession that we already postulated. So, our quest for the greatest natural number is always doomed to failure: there will never be such thing as the number that is larger than all others.
When we talk about the endless sequence of natural numbers, we say: "1, 2, 3, ... to infinity," as if the infinite were a place, or a limit which cannot go beyond, or the representation of the greatest natural number we can imagine.
But the infinite is much more than the definition that the Webster Dictionary gives us . It is not an "inconceivably great " natural number or an "Inexhaustible quantity" because infinity is not a "value" or a "number":
infinity is the sequence itself.
If the infinite is the sequence itself, then from that point of view,
infinity starts at the number 1.
This is because the number 1 is not the successor of another natural number.
If the sequence of natural numbers is the infinite, then there is no way to show as a mathematical proof that natural numbers are infinite, simply because we can not demonstrate what we accept as true by prior definition. Any other mathematical proof that deals with infinity is implicitly assuming that the infinite is the sequence of all successors of the number one, no matter if that proof is related with the natural numbers or not.
This is the case with the demonstration of Georg Cantor (1845-1918) that fractions are numerable, that is, there are as many natural numbers as fractions. Cantor devised an ingenious demonstration that proved that fractions are "countable", i.e., there are no more fractions than natural numbers. What he did was to establish a one-on-one (1-1) correspondence among the set of all fractions and the set of all natural numbers. It had to be this way because there is no other way of demonstrating that a series or a set of elements is infinite without establishing a correspondence between this set and the natural numbers.
There is a figure —shown at right—in which the fractions are aligned diagonally, which was the technique used by Cantor to show that fractions are "infinitely countable", i.e., that contrary to which the intuition tell us, there are no more fractions than natural numbers. When the fractions are arranged or composed as we see in the picture, we can cover them all of them by following the path given by the arrows. In this way, we can enumerate all the fractions as if they were "the first fraction, the second fraction, the third fraction", etc.. As the fractions are numerable, then they are the same kind of infinite as the positive integers.
The question that remains is how can these two sets be equipollent if the natural numbers enjoy the the property of natural succession, and the fractions not. That is, to every natural number always follows one, and only one, natural number, but given any fraction we cannot say which is the fraction that follows. On the other hand, between any two fractions there is always a fraction that is "in the middle" of the two (just find the arithmetic average of them), but between two integers there is not always another integer "in the middle" between them. From this we can infer that when infinite sets are involved, the properties of both series are not always necessarily mutually shared.
Galileo Galilei (1564-1642) foresaw that when we refer to infinity, we can not think of it with the same attributes that are used in the finite cases. For example, he reasoned that there is the same amount of square numbers as natural numbers (see illustration). So, the concepts "greater than" and "less than" does not apply when we handle infinite "quantities". Note by the diagram that the square of the numbers are becoming more scarce, i.e., they are increasingly separated from one another, however, the natural numbers are always separated by a unit distance. This strange behavior is known as "the paradox of Galileo." What this paradox implies is that we can not say that there are more naturals than squares, we can only assert that there is a correspondence one-on-one (1-1) among them.
To elaborate on this and other myths about the infinite, and for free books to download, see the article: Some myths about the infinite.
Another of the Cantor's successes was in demonstrating that the real numbers —the set of all possible decimals— are innumerable, that is, there is no way to establish (1-1) correspondence between the natural numbers and decimals. This demonstration is not difficult to follow, but we'll omit it to avoid filling this article with mathematical symbols.
The symbol of infinity
How can we symbolize something that we can not even imagine? We symbolize the unit with the numeral 1, the plurality by 2, or 3, etc. But infinity is not a number, for this reason it has no numeral assigned. Therefore, nothing prohibits us from using a symbol as an abbreviation when we refer to infinity.
The symbol of infinity has always been used in the Tarot cards, in the image of the Magician, to symbolize the implementation of the impossible, and the unlimited powers of the magic that transcends natural laws. In early versions of these cards, specifically, in the Tarot of Marseilles, close to 1500, the magician Arcane appears with a mysterious wizard's hat very similar to the infinity symbol ∞.
Loosely speaking, infinity is often informally symbolized by ∞ , but in the theory of set that Cantor developed the formal symbol for infinity is 
0 when we talk about the "smaller", of the infinites which is the infinitude of the natural numbers. The symbol 0 is usually read as "aleph sub 0" to distinguish it from other infinitudes that "follow" this first infinity.
For a history of the origin of the symbol ∞, see Wikipedia: Infinity : where it says:
John Wallis is usually credited with introducing ∞ as a symbol ∞ for infinity in 1655 in his De sectionibus conicis.
The reader can also see other related notes at: The History of Mathematical Symbols.
The infinite is ... infinite
Anyone would say that it can not be otherwise, but by the works of Georg Cantor, the founder of the Theory of Sets, now we know that there are several types of infinites. What we have seen so far is only the "smallest" and "easiest" from all of them!.
Cantor established "categories" to infinity to differentiate them. He used the first letter of the Hebrew alphabet, , known as "Aleph (Aleph), with subscripts to establish the sequence of the infinite. The first infinity is denoted by 0, 1 is the next, and so on.
We mentioned that the infinity of the natural numbers is 0, then, what is infinity 1? To get near to what Cantor speculated what the "next" infinity is, we need to invoke the concept of subsets of sets. Intuitively we know that a set is any collection or grouping of objects be them concrete or imaginary. For example we can imagine the set of all numbers ending in 3, and (3, 13, 23, 33, ...}, or the set of all the houses that have one door painted in white.
In contrast, a subset is formed by any set of some or all the elements of the given set. For example if we have all the first 4 natural numbers: (1, 2, 3, 4), some of the subsets of this set are the subsets {2, 3}, {1, 3}, { 3.4}, {2, 3, 4}.
The number of subsets that can be obtained from a set of n elements is 2n. That is, if the set A = {a, b} has 2 elements, then they are leaving 22 subsets {a}, {b}, Ř and {a, b}. The set Ř, the empty set, is considered as a subset of any set, and in turn, every set is considered subset of itself. The set of all subsets of any set is called the power set.
It can be shown that the amount of subsets that can be obtained from a set is always greater than the number of elements in the set in question. This was the key to Cantor to show that the quantity of real numbers in the real number line is "greater" than the amount of natural numbers in the same line. Anyone would say: "Sure, that's logical: between two integers there is always an infinite number of points." But when we go to the detail, that same can be said of fractions: "There is always an infinite number of fractions between two integers." However, it is easy to show that the fractions are numbered (that we saw earlier), while the real numbers are not.
Reviewing
0 = {1, 2, 3, ...} , and
1 = {..., 128.57, ... 0.2675, ..., 3.6719887, ..., 235.999311, ...}.
Note the reader how easy it is to conceptualize the "first" infinite, and how difficult it is to conceptualize the "next" infinite, because it contains all possible integers, and decimals at once (the fractions and the whole numbers can be expressed as decimals).
The infinites that follow the infinity 0 are given a special name: they are called transfinite. 1, is then, the first transfinite.
According to Cantor, series of the infinite is:
0, 1, 2, etc.
Some of the implications of this are that
0 + 1 = 0, 0 + 0 = 0, etc.
0 + 1 = 1, 1 + 1 = 1, etc.
The reader must internalize these expressions as concepts rather than as equations because if we take the term literally 0 + 1 = 0 and subtract 0 from both sides of the "equation", we will come the conclusion that 1 = 0 which is clearly false (this is how some of the paradoxes in mathematics are built).
More information about the transfinite numbers can be found in the article Transfinite number in Wikipedia.
Have we seen it all?
The concept of infinity is very elusive to our finite minds. As an example of this, we cannot bear in our minds all numbers under 100, but we cannot bear in our minds all the numbers greater than 100.
However, no matter how much we try, we will always have a limited view of everything that is unlimited, but that will not stop us to be bold enough to continue scrutinizing these borders of the "infinite."
Infinity has been explored by several authors for many centuries, each contributing a little to the understanding of this field. However, there is an author who ─I believe─ has faced the vertigo of infinity like no other: he is the Argentine writer Jorge Luis Borges (1899-1986).
Nobody understands the infinite better than Borges
Jorge Luis Jorge Luis Borges is the author of the marvelous short story titled The Book of Sand. The story is about a book he buys, but that can never be opened on the same page, because the book is "infinite." He can never find the first page, never the last page, never the same figures. There is never a way of opening the book on the page where it was bookmarked the previous day.
Here are a few short passages of this short history. The reader can see in them why his story is so fascinating.
I live alone in a fourth-floor apartment on Belgrano street, in Buenos Aires. Late one evening, a few months back, I heard a knock at my door. I opened it and a stranger stood there. He was a tall man, with nondescript features ─or perhaps it was my myopia that made them seem that way. .... I saw at once that he a foreigner.
"I sell Bibles," he said.
"I don't only sell Bibles. I can show you a holy book ... It may interest you".
I opened it at random. The script was strange to me. The pages, which were worn and typography poor, were laid out in double columns, as in the Bible. The text was closely printed, and was ordered into versicles. In the upper corners of the pages were Arabic numbers. I noticed that one left-page bore the number (let us say) 40,514 and the facing right-hand page 999. I turned the leaf; it was numbered with eight digits.. It also bore a small illustration, like the kind used in dictionaries─an anchor drawn with pen and ink, as if by a schoolboy's clumsy hand.
This is where Borges starts playing with the infinite. When one opens a book the pages are sequential, but that's not what happened to him. At first intention what he saw was 40,514, and in the facing page 999. When he turned the page the number was an eight-digit number, let us say, 56,783,452.
"It seems to be a version of Scriptures in some Indian language, is it not?"
"No," he replied. Then, as in confiding a secret, he lowered his voice. "I acquired the book in a town out on the plain in exchange for a handful of rupees and a Bible. Its owner did not know how to read. I suspect he saw the Book of Books as a talisman. He was of the lowest caste; nobody but other untouchables could tread his shadow without contamination. He told me his book was called the Book of Sand because neither the book nor the sand has any beginning or end. "
"Neither the book nor the sand has any beginning or end." That's what we experience every time we go to a beach or to a river bank: there is no way to have on hand the same grain twice.
The stranger asked me to find the first page.
I laid my left hand on the cover and, trying to put my thumb on the flyleaf, I opened the book. It was useless. Every time I tried a number of pages came between the cover and my thumb.
"Now find the last page. "
Again I failed. In a voice that was not mine, I barely managed to stammer, "This can't be."
Still speaking in a low voice, the stranger said "I can't be, but it is. The number of pages in this book is no more or less than infinite. None is the first page, none the last. I don't know why they're numbered in this arbitrary way. Perhaps to suggest that the terms of an infinite series admit any number."
Here we have the mathematician inside Borges reasoning: An infinite series is infinite by its elements and not by where it begins. It needs not to have a beginning nor does it need have to have order. Borges' infinity is unmanageable because there is no way to establish a matching-to-one (1-1) between the pages of books and natural numbers.
I went to the bed and did not sleep. At three or four in the morning I turn on the light. I got down the impossible book and leafed through its pages. On one of them I saw engraved a mask. The upper corner of the page carried a number, which I no longer recall, elevated to the ninth power.
Borges realizes that in an infinite book the numbering will become son large that the will occupy the the whole page. That's why he uses the exponents. A figure to the ninth power could be something like, 22,676,1909. (Of course, at some instance, a number in a page could be bigger than the book itself!)
Summer came and went, and I realized that the book was monstrous. What good it did to me to think that I, who looked upon a volume with my eyes, who held it in my hands, was any less monstrous? I felt the book was was a nightmarish object, an obscene thing that affronted and tainted reality itself?
I thought of fire, but I feared that the burning of an infinite book might likewise prove infinite and suffocate the planet with smoke.
Borges implies that it is like losing the mind realizing that it is a nightmare having at hand an infinite object, and decides to get rid of this madness "monstrous" book.
I recalled reading that the best place to hide a leaf is in a forest. Before retirement, I worked on Mexico street, at the Argentine National Library, which contains nine hundred thousand volumes. I knew that to the right of the entrance a curved staircase leads down into the basement, where books and maps and periodicals are kept. One day I went there and, slipping past a member of the staff and trying not to notice at what height or distance from the door, I lost the Book of Sand on one the basement's musty shelves.
Borges certainly will try not to go through the street where the library is, but he will be for the rest of his life trying to decipher what an infinite book can contain? Is it that he had in his hands an infinite dictionary? I think so.
Georg Cantor discovered and explored for posterity many unexpected properties and characteristics of the mathematical infinity that even today are still debated. For that he paid with his mental health.
Almost simultaneously, in the other hemisphere of the world, Jorge Luis Borges, with a graceful literary intuition, was also writing about the infinite, but from another standpoint of view: the infinite as a non-static, indomitable, insurmountable entity.
Cantor stated the infinity in theorems; Borges challenged the reader to recreate and imagine in their minds their own infinite.
Notes.
In a separate series of articles devoted exclusively to the Book of Sand by Borges I try to show that more than an infinite book, the kind of book that Borges had in his hands was a transfinite book. The interested reader can visit:
* Nobody understands the infinite so well as Borges. We have seen the infinite generated starting with the number 1. But Borges gives an interesting twist to this: Why infinity must have a beginning? If the requirement for an infinite is not having an end, it can also be infinite without having no beginning.
Why to be infinite a sequence of integers has to be evenly space as in many mathematical series? That is what is implicit in the monstrous "Book of Sand."
* The Book of Sand of Borges and the Continuum of Cantor. It was Aristotle, Pascal, Galileo and others who made the first approaches to the understanding of the infinite. In modern tines, it was Borges with his literature, and Cantor with his theorems that have contributed so much to deepen and refine our ideas of this deep concept.
* The Book of Sand is a transfinite book. The fact that Borges' book never opened in the same page, and never ended with the same page, means that each time that numbering was random. This article it a mathematical attempt to show that this must be a transfinite book.
* Is the Book of Sand a book from the fourth dimension?. In the study of the fourth dimension is where we find the concepts of hyperspace, hypercube, hyperplane, etc.. The objects that surround us every day would be three-dimensional sections of the hyperspace. But from the qualities that Borges presents from his "Bible" we can infer that the Book of Sand is also three-dimensional cut of a four-dimensional book, that is, it is like a book from the fourth dimension manifested in our three dimensions each time it is opened.
E. Pérez
jul-10
Notes:
See also in this Website the following related articles:
Three unexpected behaviors of the infinite ,
Some myths about the infinite ,
Some myths about the infinite - Part 2 .
References in print:
[1] Borges, J. L. (1975). El libro de arena. Emecé Editores, S. A., Buenos Aires.
[2] Borges, J. L. (1978). The Book of Sand. Trans. by Di Giovanni, N. Dutton Press, New York.
[3] Maor, E. (1991). To Infinity and Beyond: A Cultural History of the Infinite. Princeton University Press. New Jersey.
References on line
